 Open Journal of Applied Sciences, 2013, 3, 99-101 doi:10.4236/ojapps.2013.31B1020 Published Online April 2013 (http://www.scirp.org/journal/ojapps) Rogue Waves of the Kundu-DNLS Equation Shibao Shan, Chuanzhong Li, Jingsong He Department of Mathematics, Ningbo University, Ningbo, China Email: piaoxiaodan@163.com Received 2013 ABSTRACT In this paper, we give the Lax pair and construct the Darboux transformation of the Kundu-DNLS equation. Further-more, the rogue wave solutions of the Kundu-DNLS equation are derived by using the Taylor expansion of the breather solution. What's more, the triangular and the circular patterns of the third rouge solution are displayed. Keywords: Kundu-DNLS Equation; Darboux Transformation; Rogue Waves 1. Introduction As one of the most important integrable systems in many branches of physics and applied mathematics, the deriva-tive nonlinear Schr\"{o}dinger (DNLS) equation 02 xxxt qqiqiq (1) has been studied in optics, water wave and so on [1,2]. Considering the significance of the higher order nonlin- earities in physical system, the DNLS equation yields an integrable higher nonlinear equation, i.e. Kundu-DNLS equation [3,4] 0)2()()(*22*2QQiQQiQQiQiQxxxxxtxxxt (2) by means of a nonlinear transformation of the field with arbitrary gauge function iqeQq  . For example, setting xdxQ 2, Kundu-DNLS equation implies Eckhaus-Kundu (EK) equation. Here denotes the complex conjugate of Q, and α s a real parameter. *QRogue waves are one of those fascinating destructive phenomena in nature that have not been fully explained so far. Understanding the initial conditions that foster rogue waves could be useful both in attempts to avoid them by seafarers and in generating highly energetic pulses in optical fibers. There are several method to solve the integrable equations, for instance, Hirota method , inverse scattering transformation , bilin-ear method , Darboux transformation . In this paper, we have given the rogue waves of the DNLS and the coupled system of Hirota and Maxwell-Bloch equations [2,9,10]. Inspired by the importance of these recent in-teresting developments about the analysis of rogue waves of the NLS-type equations, we shall construct the rogue wave solutions of the Kundu-DNLS equation with the help of the Darboux transformation. 2. Darboux Transformation and Lax Pair The Darboux transformation is a powerful method used to generate the soliton solutions for integrable equations. Inspired by classical Darboux transformation for the DNLS equation, we consider the coupled Kundu-DNLS equa-tion, 0)2()()(222RQiQQiRQiQiQxxxxxtxxxt (3) 0)2()()(222QRiRRiQRiRiRxxxxxtxxxt (4) where  is a arbitrary gauge function. This form of the equation is very extensive, which is reduced to the equa-tion (2) for with the sign of the nonlinear term changed. The Kundu-DNLS equation can be obtained if α is a real parameter. *QR We first present a general framework for deriving the required conservation rule for the DNLS equation. We start with the linear set of Lax equations: ,, VU tx (5) where U and V depend on the complex constant eigen-value parameter λ. 00100148,0Re0210014*242GGiQRiVQeiiU ii, with .2)(24*2ixiixieQQiQeeQiQeG  Copyright © 2013 SciRes. OJAppS S. B. SHAN ET AL. 100 where λ is the eigenvalue, Φ is the eigenfunction corre-sponding to λ. In general, considering the universality of Darboux transformation, according to the Kundu-DNLS equation (3) and (4) , we can start from ,0000111122222dcbadcbadcbaT (6) where are functions of 222211110000,,,,,,,,,,, dcbadcbadcba.t ,x From ,1TUTUTx (7) and ,1TVTVTt (8) then the now solutions 11,RQ are given by .,2211211221211121122212112211iieQReQQ (9) Here .,,,222211212222211121222211211222211111 So far, we discussed about the determinant construc-tion of one-fold Darboux transformation of Kundu- DNLS equation. As an application of these transforma-tions of Kundu-DNLS equation, rogue wave solutions will be constructed in the next section. 3. Rogue Wave Solutions In this section, we construct the rogue wave solution of Kundu-DNLS equation . This kind of solution only ap-pears in some special region of time and space and then drown into a fixed non-vanishing plane. By making use of the Taylor expansion for the breather solution, one order rogue wave solution of for the Kundu-DNLS equation is obtained 1rQ ,2)2(11vevQtxir (10) ,8812484488888323344222221tititixixxtxtxtixtitxxv .44488888481443222222ixxtixxttxxititxititv The picture of one order rogue wave solution of the Kundu-DNLS equation and its corresponding density graph are plotted in Figure 1. When we take ,1,1,1,2 ca the picture of second rogue wave solution for the Kundu-DNLS equa-tion is displayed in Figure 2. Next, we examine third-order rogue waves. In this case, form the figures, We can get third-order rogue wave solution with the help of ,1,1,2ca the picture of third-order rogue wave solution is displayed in Figure 3. Figure 1. The One order rogue wave solution of the Kundu- DNLS equation with a = –2, c = 1, ξ = 1, η = 1. Figure 2. The second order rogue wave solution of the Kundu- DNLS equation with a = –2, c = 1, ξ = 1, η = 0.8. Figure 3. The third order rogue wave solution |Qr3|2 f the Kundu-DNLS equation with a = –2, c = 1, ξ = 0.8, η = 0.8. Copyright © 2013 SciRes. OJAppS S. B. SHAN ET AL. Copyright © 2013 SciRes. OJAppS 101 (a) (b) Figure 4. The third order rogue wave solution |Qr3|2 of the Kundu-DNLS equation w ith (a) a = –2, c = 1, S0 = 0, S1 = 500, S2 = 0; (b) a = –2, c = 1, S0 = 0, S1 = 0, S2 = 1000. We can split the third order rogue wave solution into triangle structure. A particular structure is displayed in Figure 4(a). The third-order rogue wave is seen to pos-sess a regular triangle spatial symmetry structure. What's more, we also can split the third order rogue wave solution into pentagon structure. A particular struc- ture is displayed in Figure 4(b). The third-order rogue wave exhibits a regular pentagon spatial symmetry structure. 4. Conclusions In this paper, we construct the Darboux transformation for the Kundu-DNLS equation. This Darboux transfor-mation, in particular, allows us to calculate higher order rogue wave solutions in a unified way. In this way, we can derive the higher order rogue wave solutions for Kundu-DNLS equation by making use of the Darboux transformation. Particularly, these rogue wave solutions possess several free parameters. With the help of these parameters, these rogue waves constitute some patterns, such as fundamental pattern, triangular pattern, circular pattern. 5. Acknowledgements This work is supported by the NSF of China under Grant No.11271210 and K. C. Wong Magna Fund in Ningbo University. Jingsong He is also supported by Natural Science Foundation of Ningbo under Grant No. 2011A610179. Chuanzhong Li is supported by the Na-tional Natural Science Foundation of China under Grant No.11201251, the Natural Science Foundation of Zheji-ang Province under Grant No. LY12A01007. We thank Prof. Yishen Li (USTC, Hefei, China) for his long time support and useful suggestions. 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